Rescaling the Pythagorean Theorem

The Pythagorean theorem can apply to any shape, not just triangles. It can measure nearly any type of distance. And yet this 2000-year-old formula is still showing us new tricks.

This is a companion discussion topic for the original entry at

there is another explanation with the Thales Theorem (or maybe triangle intersection theorem in USA?).

When you have a line which cuts the two sides of a triangle and that is parallel to the third side then you have the same ratio everywhere :
c/sqrt(1+(b/a)²)= a/1 = b/(b/a)

(forgive my english it’s not my natural language … math neither !)

I forgetted to say that the Thales Theorem applies to every triangles and not only the square ones as the Pythagorean one does.

Hi Sylvain, thanks for the comment! I hadn’t heard of Thales theorem before, but it’s a nice result – perhaps a topic for an upcoming article :).

Hi Kalid,

In France it’s called the Thales’ Theorem but in USA I think it’s called the Intercept Theorem and you can find details on

The story behind this is that Thales was the first able to calculate the Gizeh Pyramid height because he though there was the same relationship between his shadow and his height and the pyramid shadow and pyramid height, he considered that the sun rays were parallels so that the same ratio applied between his values and the pyramid ones. Nice story.

Oups it was not Gizeh but Cheops pyramid and all the story is told on wikipedia.

How do you find the slope when you have the Pitch and Rise/Run?

@Lisa: Actually, slope is the same thing as rise/run.

your appendix was helpful too :slight_smile:

If alone these were available some fifty to sixty years back! then I would not have been terrorised by mathematics.


When the Pythagorean Theorem was rearranged , you exposed the slope,
Amazing to me. THIS is the kind of essence I’ve been searching for in
mathematics: Thank you Kalid.

started reading articles on this site recently… but now i’m addicted… and revising my basics and improving understanding the math